The Gist: A college in Kentucky admitted both black and white students without discrimination. The Kentucky government passed a law prohibiting the teaching of black and white students at the same school. The U.S. Supreme Court upheld the law.
The Details: Berea College was a private school chartered as a corporation, and the only desegregated school in Kentucky (hence, it is clear the law passed by the Kentucky legislature was solely and directly aimed at ending the practice at this specific school). The law passed by the Kentucky government was called "An Act to Prohibit White and Colored Persons from Attending the Same School." The law not only prohibited corporations from teaching white and black students in the same school, but "groups" and "individuals" as well.
The 14th amendment states: No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws. Berea College argued the Kentucky law violated their 14th amendment rights as citizens and deprived it of its rights without due process of law.
Instead of focusing on the 14th amendment, the Court instead ruled that because the school was a corporation chartered under Kentucky law, the Kentucky legislature had the right to enforce the segregation law against the school. Without really touching upon the 14th amendment, the court nevertheless expanded its prior rulings in Plessy and Cumming by allowing states to enforce segregation laws on private and public institutions alike. Once again, the Court succeeded in gutting the true meaning of the 14th amendment and increasing the power of local governments to do what they wanted without federal interference.
This case was later overruled by Brown v. Board of Education (1954).